Continuity Definition and 876 Threads

Homework Statement Been awhile since I looked at this, just seeing if I still know what I'm doing here. Suppose : f(x,y) = \frac{x^{2/3}y^2}{x^2 + |y|^3} for (x,y) ≠ (0,0). 1. Show that on every straight line through the origin the limit as (x,y) → (0,0) of f(x,y) exists. 2. Does the...

Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations I have no idea. The Attempt at a Solution I seriously have no idea. This is for my analysis course, and I'm not...

Hi! I think I've managed to solve this problem, but I'd like it to be checked Homework Statement show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative: $$f_{+}'(x_0),$$ and left derivative $$f_{-}'(x_0)$$ in $$x_0\in A$$, then $$f$$ is continuos in $$x_0.$$...

I used Wolfram Alpha to evaluate: lim tan[(2nπ)/(1 + 8n)] n->infinity it says that it can use the continuity of tan(n) at n = π / 4 to rewrite the aforementioned function as: tan[lim ((2nπ)/(1 + 8n))] n->infinity What is it talking about? I was taught to use certain properties of trig...

Homework Statement f(x) = [x^2]sinπx, x ∈ R, being [x] the integer part of x Homework Equations The Attempt at a Solution I'd say it's trivially continuos on all R, because it's the product of two continuos functions: y: = [x^2] and g: = sinπx. However, being part of my...

Homework Statement Hi everyone, I'm kind of struggling with determining continuity of functions.Homework Equations The Attempt at a Solution For example, f(x)=|x-1| I know is continuous on ℝ but how do I show this clearly. I my head, I just think that the function is valid for any value of x...

Hello, My question is on the Klein-Gordon equation and it's relation to the continuity equation, so for a Klein-Gordon equation & continuity equation of the following form, I have attained the following probability density and probability current relations (although not normalised correctly...

hi everyone, I've found this exercise on a textbook and it doesn't resemble any exercise I've seen before. I just want to know how to proceed, you don't have to solve it for me :) Homework Statement Study the continuity of the following functions, defined by: 1- f(x) = lim...

what happens to the continuity of wave function! In the presence of a delta potential, how does the continuity of the wave function gets violated?

Homework Statement I am a student in advanced calculus and I am having an issue with a grade I just received. the question was as follows: If a function f:[0, 3]→ℝ is continuous use the Archimedes Riemann theorem to show that f is also integrable. I want to take my answer to my...

Homework Statement Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c]. Homework Equations The Attempt at a Solution This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even...

Homework Statement The continuity equation provides a second relation between the vA and vB, this time in terms of the diameters d and D. Numerical check: If the diameters are d = 1 cm and D = 10 cm, what is the ratio of the speeds, vB/vA? Homework Equations To clarify, is both d and D...

Homework Statement Hi guys, I'm trying to prove that matrix inversion is continuous. In other words, I'm trying to show that in a normed vector space the map \varphi: GL(n,R) \to GL(n,R) defined by \varphi(A) = A^{-1} is continuous.Homework Equations The norm that we're working in the...

Homework Statement To solve a problem in a book, I need to know whether or not the following is true: Let f be a real-valued, decreasing differentiable function defined on the interval [1, \infty) such that \lim_{x \rightarrow \infty} f(x) = 0. Then the derivative of f is continuous...

Homework Statement Given the function: x*y / (4-x²-2y²) if x²+2y² ≠4 0 if x²+2y² = 4 Check if the function is continuous. Homework Equations The Attempt at a Solution I tried using various ways to see if the result of the limit as (x,y)→(2,0) was the same...

Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations The Attempt at a Solution I really don't know how to do this. Wikipedia entry for 'base sets' redirects to Pokemon...

Homework Statement http://i.imgur.com/69BmR.jpg Homework Equations The Attempt at a Solution a, c are right because f(c) is continuous. b, d are right because f'(c) is differentiable over the interval I am not sure about e. Can anyone explain to me?

1.The Question The function f(x)= x2/x if (x≠0) 0 if(x=0) The Attempt at a SolutionI thought this had a removable...

Homework Statement 1- Let f be a continuous function for all real numbers such that : \lim_{x\rightarrow+\infty}f(x)=L and \lim_{x\rightarrow-\infty}f(x)=L' and that LL'≤0. Prove that f equals 0 at some point C in ℝ. 2- Let f be a continuous function on [a,b] such that for every...

Hello MHB, the f(0) of this function doesn't exist, so I am i wrong or this question don't hv solution?

Will anyone give me the derivations for continuty equation of fluid and euler's equation of fluid motion .

I am trying to complete a previous exam and have come across a question which I am unable to do. I know how to complete an epsilon delta proof for limits, however, not to prove continuity... We haven't seemed to cover this in our lecture notes :/ Using an epsilon-delta technique, prove that...

Could someone explain this to me in terms of limits and derivatives instead of plain english? For example, how would you solve a question that says find whether the function f is differentiable at x=n and a question that asks find whether the function f is continuous at = n...

Homework Statement Let f be q function defined from [a,b] to [a,b] such that for every (x,t) in [a,b]^2: l f(x)-f(t) l < l x-t l 1- prove that f is continuous on [a;b] 2-prove that f accepts a steadfast point in [a,b] The Attempt at a Solution Should i try to use the definition of a limit...

Homework Statement Let f be a continuous function on the interval I=[a,b] such that for every x in [a,b] f(x)≠0. Show that the function f(x) doesn't change its sign.( like increasing or decreasing) The Attempt at a Solution Well for this to be true, we need to have f(a)>0 and f(b)>0 and f(x)...

Homework Statement Let zn = Arg(-1 + i/n). Find limn→∞ zn Homework Equations Definition of convergence of a sequence. The Attempt at a Solution Well zn = Arg(-1 + i/n) = arctan(-1/n). So it seems clear that limn→∞arctan(-1/n) = arctan(limn→∞ -1/n) = arctan(0) = \pi. Which is true if...

Homework Statement lim of (y^2)(sin^2x) /(x^4+y^4) as (x,y) approaches (0,0) Homework Equations The Attempt at a Solution I got the limit as (x,y) approaches (0,y) and as (x,y) approaches (x,0), and it equals 0. But now I'm unsure of what to to next. I think it was the limit...

Homework Statement Let g be a function defined as g(x)=(\frac{1}{4})x^{2}-sin(x) Give a monotony table for g in the domain [-\frac{\pi}{2},\frac{\pi}{2}] The Attempt at a Solution I calculated the first derivative of g and i got g'(x)=(1/2)x-cos(x) and then when I wanted to...

Homework Statement Please click on the link for the question. http://i1154.photobucket.com/albums/p526/cathy446/physicsquestion_zps49e16ab1.jpg Assume that air spreads out after coming out from the tube at 2. The speed over tube 1 is almost zero. Homework Equations Knowledge problem on...

Homework Statement Let n be a natural number. Prove that the equation: x^{2}(cos(x))^{n}+xsin(x)+1=0 has an infinite amount of solutions. The Attempt at a Solution I named that equation f(x)=0 and I said that f(a)<f(0)<f(b) and that f(a) x f(b) < 0. Should I choose n=1 and...

Homework Statement Show that if (x_{n}) is a sequence in a metric space (E,d) which converges to some x\inE, then (f(x_{n})) is a convergent sequence in the reals (for its usual metric). Homework Equations Since (x_{n}) converges to x, for all ε>0, there exists N such that for all...

Suppose I have two charged particles with charge densities ρ1(r,t) and ρ2 (r,t) with corresponding velocity fields V1(r,t) and V2(r,t). Can I write continuity equation for the combined system? Wouldn't charges moving with different velocities would contribute differently to the current which...

Homework Statement Define functions f and g on [-1,1] by f(x) = xcos(1/x) if x≠0 and 0 if x = 0 g(x)= cos(1/x) if x≠0 and 0 if x = 0 (These are piecewise defined. I don't know how to type them in here.) Prove that f is continuous at 0 and that g is not continuous at 0. Explain why...

http://chutzpah.typepad.com/.a/6a00e55180ed5c88340120a75cf644970b-pi How do the circles still intersect at the bottom, and at 2 points like the top 2 circles?

Hi all, I have the following question: Suppose f: [0, ∞) \rightarrow ℝ and f is concave , nondecreasing and bounded on [ 0, ∞) . Does it follow that f is continuous on [ 0, ∞) ? Thanks in advance, H.

Homework Statement Follow the link to see the question, http://img507.imageshack.us/img507/2246/fluidquestion.png Homework Equations The Attempt at a Solution currently I can't do part a) but from using part a) I can obtain the forces acting on the cone by using the first...

Homework Statement #1. If limit[x->a]f(x) exists, but limit[x->a]g(x) doesnt, limit[x->a](f(x)+g(x)) doesn't exist. T/F? (Proof or example please) #2. prove that if f is continuous, then so is |f| #3. f(x) = [[x]]+[[-x]] for what a does limit[x->a]f(x) exist? Where is f discontinuous...

Hello everybody! I am using in my studies this beautiful book by Kippenhahn & Weigert, "Stellar Structure and Evolution", but I have some problems about collapsing polytropes (chapter 19.11)... After defining dimensionless lenght-scale z by: r=a(t)z and a velocity potential \psi...

First of all, thanks to all the PF mentors out here, especially TSny and pgardn, who have made physics doable and are helping me accomplish my dreams! Even when putting in the work its not easy to get all this stuff! Homework Statement A large water tank has an inlet pipe and an outlet...

Hi, All: I am working on a proof of the fact that any two norms on a f.dim. normed space V are equivalent. The idea seems clear, except for a statement that (paraphrase) any norm in V is a continuous function of any other norm. For the sake of context, the whole proof goes like this...

Analysis Question--differentiabillity, continuity Homework Statement Suppose f:\mathbb{R}\to\mathbb{R} is a C^\infty function which satisfies the equation f''(x)=-x^2f(x) along with f(0)=1, f'(0)=0. Prove that there is an a>0 such that f(a)=0. Do not use any results from differential...

Hey everyone, I was just curious about the nature of the cube root function f(x)=x^{1/3}. I know that its derivative is obviously \frac{1}{3}x^{-2/3} which has a discontinuity at x=0. However, in the non-mathematical sense, the graph of y=f(x) looks smooth - I don't see any angles or cusps like...

Homework Statement If f is integrable on [a,b], prove that there exists an infinite number of points in [a,b] such that f is continuous at those points. Homework Equations I'm using Spivak's Calculus. There are two criteria for integrability that could be used in this proof (obviously...

hello i have question : how the superposition principle proves continuity of ψ in potential barrier

Hi guys, I just started reading an introductory book on analysis. I'm up to the part where they talk about functions now, and I'm getting lost. The theorem that I'm having trouble envisioning is: Let f: D-> R and let c be an accumulation point of D. Then limx->cf(x)=L iff for each...

Homework Statement Give values of x where f(x)= x-1 / x2+4x+3 is continuous f(x)= x2-4 / x2+x-2 Where x has removable discontinuity Homework Equations The Attempt at a Solution Continuous, jump, infinite, removable. It's been so long that I do not remember. I tried to...

Why is partial derivative with respect to time used in the continuity equation, \frac{\partial \rho}{\partial t} = - \nabla \vec{j} If this equation is really derived from the equation, \frac{dq}{dt} = - \int\int \vec{j} \cdot d\vec{a} Then should it be a total derivative with...

Homework Statement If the set \Z of integers is equipped with the relative topology inherited from ℝ, and κ:\Z→\Z_n (where κ is a canonical map and \Z_n is the residue class modulo n) what topology/topologies on \Z_n will render κ globally continuous? Homework Equations The Attempt...

Hello, I was reading through some lecture notes on Single-Variable Calculus, and the teacher gave this definition of continuity: "A function f is called continuous at a point p if a value f(p) can be found such that f(x) → f(p) for x → p. A function f is called continuous on [a, b] if it is...

Let f : R to R be a continuous function, and suppose that definite integral from m to n |∫(m to n)f(x)dx|≤(n-m)^2 for every closed bounded interval [m, n] in R. Then is it the case that f(x) = 0 for all x in R? I tried using fundamental theorem of calculus but got stuck, since I only got that...